Electrical Engineering 20N - Fall 1999 - Final Exam

Electrical Engineering 20N - Fall 1999 - Final Exam - EECS...

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Unformatted text preview: EECS 20. Solutions to Final Exam. December 16, 1999. 1. 20 points A signal is mathematically described as a function. As we have seen in the course, these may be functions of time and space or they may be sequences. For example, a one-second long soundwave may be described as a function Sound : [0 , 1] → AirPressure . Propose mathematical descriptionss for the signals corresponding to the following intuitive descriptions. Give a brief justification for your answer. (a) 5 The signal obtained after sampling Sound (described above) 8,000 times. (b) 5 A black-and-white 800 × 600 pixel image. (c) 5 A one-second long black-and-white video with 30 frames per second. (d) 5 The sequence of buttons you press with your TV remote control. Answer (a) We have SampledSound : { , 1 , ··· , 8 , 000 } → AirPressure , where for all n , SampledSound ( n ) = Sound ( n/ 8 , 000). (b) We have Image : { , ··· , 799 } × { , ··· , 599 } → { , ··· , 255 } , where the range is an 8-bit color-map index. (c) We have Video : { , ··· , 29 } → Images , where Images are all functions of the form Image . (d) The sequence could be modeled as Ints → { 1 , ··· , 5 , open , close } , assuming that the available buttons are: 1 , ··· , 5 , open , close . 2. 20 points (a) 5 Write cos 3 ( ωt ) in terms of cosines and sines of multiples of ωt . (b) 5 Express in polar form all the distinct roots of the equation z 5 = 2. (c) 5 Find A and θ so that A sin( ωt + θ ) = sin( ωt ) + cos( ωt ) . (d) 5 Express the fraction 1+ jω 1 − jω in rectangular and polar forms. 1 x t-1 1 1 Figure 1: The graph of x for problem 3 Answer (a) Write cos 3 ( ωt ) = 1 / 8[ e iωt + e − iωt ] 3 = 1 / 8[ e 3 iωt + e − 3 iωt + 3 e iωt + 3 e − ωt ] = 1 / 4[cos(3 ωt ) + 3cos( ωt ) . ] (b) The distinct roots are: { 2 1 / 5 × e i 2 πk/ 5 | k = 0 , 1 , 2 , 3 , 4 } . (c) Since sin( ωt + θ ) = sin( ωt )cos( θ ) + cos( ωt )sin( θ ) = sin( ωt ) + cos( ωt ) , we must have A cos( θ ) = A sin( θ ) = 1 which gives A = √ 2 ,θ = π/ 4 . (d) We have 1 + jω 1 − jω = (1 + jω ) 2 1 + ω 2 = 1 − ω 2 1 + ω 2 + j 2 ω 1 + ω 2 , and 1 + jω 1 − jω = 1 × e j 2 tan- 1 ω ....
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This note was uploaded on 05/17/2009 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at University of California, Berkeley.

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Electrical Engineering 20N - Fall 1999 - Final Exam - EECS...

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